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Q1 Number Theory Properties of Integer Sequences and Digit Analysis View

What is the representation in base 2 of the number $(15)_8$ given in base 8?
A) $(1001)_2$
B) $(1011)_2$
C) $(1101)_2$
D) $(1110)_2$
E) $(1111)_2$

Q2 Indices and Surds Numerical Arithmetic with Fractions and Decimals View

$$\frac { 16 ^ { 3 } } { 24 ^ { 3 } + 16 ^ { 3 } + 8 ^ { 3 } }$$
What is the result of this operation?
A) $\frac { 1 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 1 } { 5 }$
D) $\frac { 4 } { 7 }$
E) $\frac { 2 } { 9 }$

Q3 Exponential Equations & Modelling Evaluate Expression Given Exponential/Logarithmic Conditions View

$$\frac { 3 ^ { x } } { 2 ^ { 2 x } } = \frac { 1 } { 5 }$$
Given this, what is the value of the expression $5 ^ { \frac { 1 } { x } }$?
A) $\frac { 3 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 9 } { 4 }$
D) $\frac { 9 } { 5 }$
E) $\frac { 5 } { 6 }$

Q4 Indices and Surds Evaluating Expressions Using Index Laws View

Given that $x = \sqrt [ 4 ] { 5 }$,
$$\left( x ^ { 2 } - 2 \right) ^ { - 1 }$$
which of the following is this expression equal to?
A) $1 + \sqrt [ 4 ] { 5 }$
B) $2 + \sqrt [ 4 ] { 5 }$
C) $1 + \sqrt { 5 }$
D) $2 + \sqrt { 5 }$
E) $1 + 2 \sqrt { 5 }$

Q5 Indices and Surds Simplifying Algebraic Expressions with Indices or Factoring View

$$\frac { x ( y + z ) + z ( y - x ) } { x ^ { 2 } + x y + x z + y z }$$
Which of the following is the simplified form of this expression?
A) $\frac { x } { x + y }$
B) $\frac { y } { x + y }$
C) $\frac { z } { x + z }$
D) $\frac { y } { x + z }$
E) $\frac { y } { y + z }$

Q6 Indices and Surds Conjugate Surds and Sum Evaluation via Identities View

For positive real numbers x and y,
$$\begin{aligned} x \cdot y & = 5 \\ x ^ { 2 } + y ^ { 2 } & = 15 \end{aligned}$$
Given this, what is the value of the expression $x ^ { 3 } + y ^ { 3 }$?
A) 40
B) 45
C) 50
D) 60
E) 75

Q7 Simultaneous equations View

Let x and y be real numbers such that
$$\begin{aligned} & x ^ { 2 } - 4 y = - 7 \\ & y ^ { 2 } - 2 x = 2 \end{aligned}$$
Given this, what is the sum $x + y$?
A) 3
B) 4
C) 5
D) $\frac { 4 } { 3 }$
E) $\frac { 5 } { 3 }$

Q8 Exponential Functions Exponential Equation Solving View

Let x be a real number such that
$$( \sqrt { 7 } + \sqrt { 3 } ) ^ { x } = 4$$
Given this, which of the following is the expression $( \sqrt { 7 } - \sqrt { 3 } ) ^ { x }$ equal to?
A) $2 ^ { - x }$
B) $2 ^ { - x + 1 }$
C) $4 ^ { x }$
D) $4 ^ { x - 1 }$
E) $4 ^ { x + 1 }$

Q9 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View

The sum of all two-digit natural numbers with digit A in the units place is 504. What is A?
A) 5
B) 6
C) 7
D) 8
E) 9

Q10 Number Theory Congruence Reasoning and Parity Arguments View

$$\left. \begin{array} { l } 2 ^ { a } \cdot 3 ^ { b } \equiv 0 ( \bmod 12 ) \\ 2 ^ { b } \cdot 3 ^ { a } \equiv 0 ( \bmod 27 ) \end{array} \right\}$$
For positive integers a and b that satisfy both congruences simultaneously, what is the minimum value of the sum $a + b$?
A) 3
B) 4
C) 5
D) 6
E) 7

Q11 Number Theory Divisibility and Divisor Analysis View

For $1 < n < 50$, how many integers n are there such that the number of positive divisors is 3?
A) 2
B) 3
C) 4
D) 5
E) 7

Q12 Inequalities Identify Always-True Inequality from Options View

Let x and y be real numbers with $-1 < y < 0 < x$. Which of the following statements are always true?
I. $x + y > 0$ II. $x - y > 1$ III. $x \cdot ( y + 1 ) > 0$
A) Only I
B) Only III
C) I and II
D) I and III
E) II and III

Q13 Exponential Functions Exponential Equation Solving View

The operation $\Delta$ is defined on the set of real numbers for all real numbers a and b as
$$a \Delta b = a ^ { 2 } + 2 ^ { b }$$
Given that $2 \Delta ( 1 \Delta x ) = 12$, what is x?
A) $\frac { 1 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 1 } { 4 }$
D) 1
E) 2

Q14 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View

Let $Z$ be the set of integers. The function $f : Z \rightarrow Z$ is defined as
$$f ( x ) = \begin{cases} x - 1 , & \text{if } x < 0 \\ x + 1 , & \text{if } x \geq 0 \end{cases}$$
Accordingly,
I. f is one-to-one. II. f is onto. III. The range of f is $Z \backslash \{ 0 \}$.
Which of these statements are true?
A) Only I
B) Only II
C) Only III
D) I and II
E) I and III

Q15 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View

$$\begin{aligned} & f ( x ) = | 2 x - 5 | \\ & g ( x ) = | x + 1 | \end{aligned}$$
The functions are given. Accordingly, what is the sum of the x values that satisfy the equation $( g \circ f ) ( x ) = 3$?
A) $-3$
B) $-1$
C) 0
D) 2
E) 5

Q16 Composite & Inverse Functions Symmetry, Periodicity, and Parity from Composition Conditions View

A function f defined on the set of real numbers satisfies the inequality
$$f ( x ) < f ( x + 2 )$$
for every real number x.
Accordingly,
I. $f ( 1 ) < f ( 5 )$ II. $| f ( - 1 ) | < | f ( 1 ) |$ III. $f ( 0 ) + f ( 2 ) < 2 \cdot f ( 4 )$
Which of these statements are always true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

Q17 Proof True/False Justification View

A student made an error while proving the following claim that he thought was true.
Claim: For any sets $A$, $B$, $C$, we have $A \backslash ( B \cap C ) \subseteq ( A \backslash B ) \cap ( A \backslash C )$.
The student's proof:
If I show that every element of the set $A \backslash ( B \cap C )$ is in the set $( A \backslash B ) \cap ( A \backslash C )$, the proof is complete.
Now, let $x \in A \backslash ( B \cap C )$. (I) From this, $x \in A$ and $x \notin ( B \cap C )$. (II) From this, $x \in A$ and $( x \notin B$ and $x \notin C )$. (III) From this, $( x \in A$ and $x \notin B )$ and $( x \in A$ and $x \notin C )$. (IV) From this, $x \in A \backslash B$ and $x \in A \backslash C$. (V) From this, $x \in [ ( A \backslash B ) \cap ( A \backslash C ) ]$.
In which of the numbered steps did this student make an error?
A) I
B) II
C) III
D) IV
E) V

Q18 Solving quadratics and applications Evaluating an algebraic expression given a constraint View

Let a and b be positive integers. The sum of the coefficients of the polynomial
$$P ( x ) = ( x + a ) \cdot ( x + b )$$
is 15. What is the sum $a + b$?
A) 10
B) 9
C) 8
D) 7
E) 6

Q19 Roots of polynomials Determine coefficients or parameters from root conditions View

$$\begin{aligned} & P ( x ) = x ^ { 2 } - 2 x + m \\ & Q ( x ) = x ^ { 2 } + 3 x + n \end{aligned}$$
polynomials are given. These two polynomials have a common root and the roots of the polynomial $P(x)$ are equal, so what is the sum $m + n$?
A) $-5$
B) $-3$
C) 2
D) 4
E) 5

Q20 Discriminant and conditions for roots Condition for repeated (equal/double) roots View

$$y = x ^ { 2 } - 2 ( a + 1 ) x + a ^ { 2 } - 1$$
If the parabola is tangent to the line $y = 1$, what is a?
A) $\frac { -3 } { 2 }$
B) $\frac { -3 } { 4 }$
C) 0
D) 1
E) 2

Q21 Combinations & Selection Selection with Group/Category Constraints View

A florist has roses of 5 different colors in large quantities and 2 types of vases. A customer wants to buy a total of 3 roses of 2 different colors and 1 vase.
In how many different ways can this customer make the purchase?
A) 15
B) 20
C) 25
D) 40
E) 50

Q22 Combinations & Selection Combinatorial Probability View

A bag contains 5 red and 4 white marbles.
When 3 marbles are drawn randomly from this bag at the same time, what is the probability that there are at most 2 marbles of each color?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 5 } { 6 }$
D) $\frac { 7 } { 8 }$
E) $\frac { 8 } { 9 }$

Q23 Trig Graphs & Exact Values View

$$\frac { \cos 135 ^ { \circ } + \cos 330 ^ { \circ } } { \sin 150 ^ { \circ } }$$
What is the value of this expression?
A) $\sqrt { 3 } - \sqrt { 2 }$
B) $\sqrt { 3 } - 1$
C) $\sqrt { 2 } - 1$
D) $\sqrt { 2 } + 1$
E) $\sqrt { 2 } + \sqrt { 3 }$

Q24 Sine and Cosine Rules Find an angle using the cosine rule View

ABCD is a square, $|BE| = 5$ cm, $|EC| = 7$ cm, $m(\widehat{EAC}) = x$.
According to the given information, what is $\tan x$?
A) $\frac { 4 } { 13 }$
B) $\frac { 6 } { 13 }$
C) $\frac { 9 } { 13 }$
D) $\frac { 5 } { 17 }$
E) $\frac { 7 } { 17 }$

Q25 Quadratic trigonometric equations View

$$\cos x \cdot \cos 2x = \frac { 1 } { 16 \sin x }$$
Given this, what is $\sin 4x$?
A) $\frac { 1 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 1 } { 4 }$
D) $\frac { \sqrt { 2 } } { 2 }$
E) $\frac { \sqrt { 3 } } { 2 }$

Q26 Solving quadratics and applications Quadratic equation with parametric or self-referential conditions View

$$x ^ { 2 } - ( \sin a ) x - \frac { 1 } { 4 } \left( \cos ^ { 2 } a \right) = 0$$
One root of the equation is $\frac { 2 } { 3 }$. Accordingly, what is $\sin a$?
A) $\frac { \sqrt { 2 } } { 2 }$
B) $\frac { \sqrt { 2 } } { 3 }$
C) $\frac { \sqrt { 2 } } { 6 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 1 } { 3 }$

Q27 Complex Numbers Arithmetic Trigonometric/Polar Form and De Moivre's Theorem View

On the set of complex numbers
$$f ( z ) = 1 - 2 z ^ { 6 }$$
a function is defined. For $z _ { 0 } = \cos \left( \frac { \pi } { 3 } \right) + i \sin \left( \frac { \pi } { 3 } \right)$, what is $f \left( z _ { 0 } \right)$?
A) $1 + i$
B) $2i$
C) $1 - i$
D) $-1$
E) $3$

Q28 Complex Numbers Arithmetic Identifying Real/Imaginary Parts or Components View

$$( | z | + z ) \cdot ( | z | - \bar { z } ) = i$$
Which of the following is the imaginary part of the complex number z that satisfies the equation?
A) $\frac { 2 } { | z | }$
B) $\frac { 1 } { | z | }$
C) $\frac { - | z | } { 2 }$
D) $\frac { 1 } { 2 | z | }$
E) $- | z |$

Q29 Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View

For the complex number $z = a + ib$ whose distance to the number 1 is 2 units and whose distance to the number i is 3 units, what is the difference $a - b$?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 7 } { 2 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 7 } { 3 }$

Q30 Laws of Logarithms Solve a Logarithmic Equation View

$$\log _ { 2 } 3x + \log _ { 4 } x ^ { 2 } = 2$$
What is the value of x that satisfies the equation?
A) $\frac { \sqrt { 2 } } { 2 }$
B) $\frac { 3 \sqrt { 2 } } { 2 }$
C) $\frac { 5 \sqrt { 2 } } { 2 }$
D) $\frac { \sqrt { 3 } } { 3 }$
E) $\frac { 2 \sqrt { 3 } } { 3 }$

Q31 Exponential Equations & Modelling Evaluate Expression Given Exponential/Logarithmic Conditions View

$$\begin{aligned} & 2 ^ { x } = \frac { 1 } { 5 } \\ & 3 ^ { y } = \frac { 1 } { 4 } \end{aligned}$$
Given this, what is the value of the product $x \cdot y$?
A) $\frac { \ln 3 } { \ln 2 }$
B) $\frac { \ln 15 } { \ln 2 }$
C) $\frac { \ln 5 } { \ln 4 }$
D) $\frac { \ln 25 } { \ln 3 }$
E) $\frac { \ln 5 } { \ln 6 }$

Q32 Sequences and Series Evaluation of a Finite or Infinite Sum View

$$\sum _ { n = 4 } ^ { 9 } \left( \prod _ { k = 1 } ^ { n } \frac { k + 1 } { k } \right)$$
What is the result of this operation?
A) 45
B) 48
C) 50
D) 52
E) 54

Q33 Sequences and Series Limit Evaluation Involving Sequences View

The sequence $\left( a _ { n } \right)$
$$a _ { n } = \begin{cases} 2 ^ { n } + 1 , & n \equiv 0 ( \bmod 2 ) \\ 2 ^ { n } - 1 , & n \equiv 1 ( \bmod 2 ) \end{cases}$$
is defined in the form. Accordingly, what is the value of the expression $\frac { a _ { 9 } - a _ { 7 } } { a _ { 8 } - 4 \cdot a _ { 6 } }$?
A) $-2 ^ { 8 }$
B) $-2 ^ { 7 }$
C) $-2 ^ { 6 }$
D) $-2 ^ { 5 }$
E) $-2 ^ { 4 }$

Q34 Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View

Below, a sequence of circles drawn side by side is given. In this sequence; the radius of the first circle is 4 units and the radius of each subsequent circle is half the radius of the previous circle.
What is the sum of the circumferences of all circles in this sequence in units?
A) $15 \pi$
B) $16 \pi$
C) $18 \pi$
D) $\frac { 31 \pi } { 2 }$
E) $\frac { 33 \pi } { 2 }$

Q35 Matrices Matrix Algebra and Product Properties View

Let a, b and c be positive real numbers,
$$\left[ \begin{array} { l l } a & b \\ 0 & c \end{array} \right] \cdot \left[ \begin{array} { l l } a & b \\ 0 & c \end{array} \right] = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 4 \end{array} \right]$$
The matrix equation is given. Accordingly, what is the sum $a + b + c$?
A) $\frac { 11 } { 3 }$
B) $\frac { 7 } { 4 }$
C) 4
D) 5
E) 6

Q36 Matrices Linear System and Inverse Existence View

For a matrix A with multiplicative inverse $A^{-1}$,
$$\left[ \begin{array} { l l } 2 & 1 \end{array} \right] \cdot \left[ \begin{array} { l l } 1 & 0 \\ 3 & 1 \end{array} \right] ^ { - 1 } \cdot \left[ \begin{array} { l } 1 \\ 4 \end{array} \right] = [ a ]$$
In the matrix equation, what is a?
A) 1
B) 2
C) 3
D) 4
E) 5

Q37 Matrices Linear System and Inverse Existence View

$$\begin{aligned} & A = \left[ \begin{array} { l l } 2 & 3 \\ 1 & 2 \end{array} \right] \\ & B = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 5 \end{array} \right] \end{aligned}$$
With the matrix notation
$$( 2 A - B ) \cdot \left[ \begin{array} { l } x \\ y \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \end{array} \right]$$
Which of the following is the system of linear equations?
A) $\begin{aligned} & x - 4 y = 0 \\ & 2 x - y = 1 \end{aligned}$
B) $\begin{aligned} & x + 2 y = 0 \\ & 2 x - 3 y = 1 \end{aligned}$
C) $\begin{aligned} & 2 x + y = 1 \\ & x - y = 0 \end{aligned}$
D) $\begin{aligned} & 3 x - 2 y = 1 \\ & 2 x + y = 0 \end{aligned}$
E) $\begin{aligned} & 3 x + 4 y = 1 \\ & 2 x - y = 0 \end{aligned}$

Q38 Simultaneous equations View

$$\lim _ { x \rightarrow 0 } \frac { \sin 3 x } { 2 - \sqrt { 4 - x } }$$
What is the value of this limit?
A) 3
B) 9
C) 12
D) 15
E) 16

Q39 Applied differentiation Limit evaluation involving derivatives or asymptotic analysis View

$$\lim _ { x \rightarrow 1 ^ { + } } ( x - 1 ) \cdot \ln \left( x ^ { 2 } - 1 \right)$$
What is the value of this limit?
A) $\frac { -1 } { 2 }$
B) $-2$
C) 0
D) 1
E) 4

Q40 Applied differentiation Limit evaluation involving derivatives or asymptotic analysis View

For a function f defined on the set of real numbers
$$\begin{aligned} & \lim _ { x \rightarrow 3 ^ { + } } f ( x ) = 1 \\ & \lim _ { x \rightarrow 3 ^ { - } } f ( x ) = 2 \end{aligned}$$
Given this, what is the value of the limit $\lim _ { x \rightarrow 2 ^ { + } } \frac { f ( 2 x - 1 ) + f ( 5 - x ) } { f \left( x ^ { 2 } - 1 \right) }$?
A) $\frac { -1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) 1
D) 3
E) 4

Q41 Applied differentiation Finding parameter values from differentiability or equation constraints View

$$f ( x ) = \begin{cases} 1 , & x \leq 1 \\ x ^ { 2 } + a x + b , & 1 < x < 3 \\ 5 , & x \geq 3 \end{cases}$$
If the function is continuous on the set of real numbers, what is the difference $a - b$?
A) $-4$
B) $-1$
C) 2
D) 3
E) 5

Q42 Composite & Inverse Functions Derivative of an Inverse Function View

For functions f and g defined on the set of real numbers
$$\begin{aligned} & f ( g ( x ) ) = x ^ { 2 } + 4 x - 1 \\ & g ( x ) = x + a \\ & f ^ { \prime } ( 0 ) = 1 \end{aligned}$$
Given this, what is a?
A) $-2$
B) $\frac { -1 } { 4 }$
C) 1
D) $\frac { 3 } { 2 }$
E) 3

Q43 Composite & Inverse Functions Recover a Function from a Composition or Functional Equation View

$$f ( 2 x + 5 ) = \tan \left( \frac { \pi } { 2 } x \right)$$
For the function $f$ given by the equality, what is the value $f ^ { -1 } ( 1 )$?
A) $\frac { \pi } { 2 }$
B) $\frac { \pi } { 4 }$
C) $\pi$
D) $2 \pi$
E) $3 \pi$

Q44 Stationary points and optimisation Determine parameters from given extremum conditions View

A third-degree real-coefficient polynomial function $P(x)$ with leading coefficient 1 has two of its roots as $-5$ and $2$.
If $P(x)$ has a local extremum at the point $x = 0$, what is the third root?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 7 } { 3 }$
D) $\frac { -5 } { 2 }$
E) $\frac { -10 } { 3 }$

Q45 Applied differentiation MCQ on derivative and graph interpretation View

Below, the graph of the derivative of a function f that is defined and continuous on the set of real numbers is given.
Accordingly,
I. $f ( 2 ) - f ( 1 ) = -2$. II. The function f has a local maximum at the point $x = 0$. III. The second derivative function is defined at the point $x = 0$.
Which of the following statements are true?
A) Only I
B) Only III
C) I and II
D) II and III
E) I, II and III

Q46 Stationary points and optimisation Geometric or applied optimisation problem View

For $x > 0$; if the point $(a, b)$ on the graph of the curve $y = 6 - x^2$ is closest to the point $(0, 1)$, what is b?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 7 } { 2 }$
D) $\frac { 9 } { 2 }$
E) $\frac { 11 } { 2 }$

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